These ‘solutions’ to Hamilton's 'No Zero' investigation by 4 different children right at the start of the year demonstrate some of the skills necessary for successful investigative mathematical learning.
It can be seen from these different approaches that there is no single ‘correct’ way of approaching the set problem. Most important is that the children have all found their own ways to make a start. Some of them have visibly changed their minds as they worked, and it is crucial that they feel that it is OK to take a risk that might not work out.
Example 1
This child has clearly laid out the answer numbers before starting work, and so has run out of room for the sequence of numbers from 1 to 50 - something that could be pointed out to the child for shaping a different approach in future investigations. Nonetheless, the child has worked through the first 3 points successfully and made a reasonable stab at question 4, which requires a bit more investigation.
Example 2
This child has used shading to identify the pattern, which has been successfully extended to answer question 3. The restriction to 50 and the lack of structure of the numbers may cause problems when the child attempts question 4.
Example 3
This child has worked out the importance of an appropriate recording strategy, enabling the identification of a pattern, which in turn might help make a generalisation for subsequent parts of the investigation.
Example 4
This child demonstrates a good understanding of a number grid as a basis for organising numbers and seems to be beginning to consider the problem of zeros in the tens column. The visible pattern enables the child to answer questions 2 and 3, where the pattern holds. The child then made a conjecture for question 4, but didn't test it to discover that the generalisation does not hold.
Try the No Zero Investigation with your own class.